A Fractal Field Guide

This is a collection of limit sets rendered mostly between Jan'93 and Apr'94 using a program that I was developing with the working title Live Fractal. These pictures differ from the usual fractals you see in a few regards

the defining mathematical formula come from families much more general than the ones for Julia and Mandelbrot sets
the pictures are rendered using inverse iteration as opposed to escape time algorithms. As a result, only the limit sets themselves are shown, without the razzle dazzle of a surrounding color field.
the inverse iteration algorithm is able to handle not only the usual IFS and Julia fractals, but also the limit sets of Kleinian groups, as popularized in the book Indra's Pearls.

In the list below, the accompanying descriptions are probably vague and unhelpful, but should give an impression of the wide variety of fractal types possible.

The rendering method used is in spirit the same as that popularized by Barnsley for his IFS fractals. Many of the images have color which indicates the number of times the algorithm visited a given region. Black indicates a few hits, green some more, and red and yellow indicate many hits. Like photography, the images you get depend on exposure time and filters. The series of images presented here are more like polaroid snapshots than serious attempts at creating high quality pictures.

Quadratic Rationals (including matings)

- inverse quadratic
- inverse quadratic
- a quadratic mating (rabbit+peapod)
- a quadratic mating (rabbit+basilica)
- a quadratic self-mating (peadpod+peapod)
- a quadratic mating
- quadratic self mating
- inverse quadratic
- quadratic rational
- quadratic rational
- quadratic rational
- quadratic rational
- inverse quadratic
- inverse quadratic

Carpet Fractals

- cubic version of a carpet fractal
- a carpet fractal
- a carpet fractal
- almost a carpet fractal
- a close-up of RABBIT
- a carpet fractal closeup

Conformal Maps and Correspondences

- derived from Barnsley's foggy coastline
- generated by Gauss's Arithmetic-Geometric mean algorithm
- generated by Gauss's Arithmetic-Geometric mean algorithm
- a quadratic correspondence
- a quadratic correspondence
- a perturbation of a mapping whose symmetry group is a dihedral group
- two superimposed members of a circle inversion group
- group of Moebius transformations
- group of Moebius transformations
- a quadratic correspondence
- not a fractal, but the image of a grid under an iterated conformal map
- see CONFORM2
- not a fractal, but the image of a circle under an iterated conformal map

Kleinian Groups

- a Kleinian group whose limit set consists of circles
- a Schottky group
- a Schottky group (verging on quasi-fuchsian)
- a Schottky group
- a Schottky group, showing the 4 generating circles
- a Kleinian group
- a nearly quasi-fuchsian group
- a quasi-fuchsian group
- partial rendering of a quasifuchsian group limit set
- 10. quasi-fuchsian limit set - this is actually a nearly plane-filling curve. This picture was used in the book The Honors Class, by Ben Yandell
- 11. A zoomed out version of 10. The box shows where 10 is situated.
- 12. Same as 11, shows isometric circles
- 13. a zoomed out version of 11. Note that peripheral resolution degrades.
- 14. Zoomed out 13
- the limit set in 10-14 rendered as a curve.
- the limit set in 10-14 rendered as a curve (zoomed out)
- the limit set in 10-14 rendered as a curve(zoomed out even more)
- in the neighbourhood of the Riley slice of Schottky space
- a Schottky group with 4 generating circles shown
- an outlying Schottky group